Down the columns (fixing \(h\)  and decreasing \(\varepsilon \) ), the data for \(Est.2\)  shows that this quantity grows as \(\varepsilon \downarrow 0\) , roughly
like \(\varepsilon ^{-1}\)  for fixed \(h\) . Across the rows, for fixed \(\varepsilon \)  and \(h\downarrow 0\) , \(Est.2\)  decreases. Diagonals (necessary
if we choose \(\varepsilon \sim h\) ) show mild growth. Comparing the last row
of the table with the row of \(Est.1\)  values shows that \(Est.2\)  consistently
yields a lower estimate of ill-conditioning than \(Est.1\) . Concerning the
growth of \(Est.2\)  as \(\varepsilon \downarrow 0\)  for fixed \(h\) , since \(h,||f||\)  do not change as \(\varepsilon \)  varies, this growth is due to \(||u^{\varepsilon }||\rightarrow 0\)  as \(\varepsilon \downarrow 0\) . The P1
finite element space does not have a non-trivial divergence free subspace,
[1]}. Thus, \(\varepsilon \downarrow 0\)  forces \(||\nabla \cdot u||\rightarrow 0\)  which forces \(||u||\rightarrow 0\) . This indicates that
ill-conditioning of penalty methods with P1 elements as \(\varepsilon \downarrow 0\)  is an essential feature caused by the lack
of a divergence-free subspace. For comparison with the last row in Table 1,
the values of \(Est.1\simeq h^{-2}+\varepsilon ^{-1}h^{-2}\)  for \(\varepsilon =6.0E-8\)  are
\(\begin{array}{cccccc}\text{ }{\small 1.0E9} & \text{ }{\small 4.3E9}\text{ } & {\small 1.7E10} &{\small 6.9E10} & {\small 2.7E11} & {\small 1.1E12}\end{array}.\) 
